Toric $g$-polynomials of hook shape lattice Path Matroid Polytopes and product of simplices

TL;DR

This paper provides explicit formulas for the $f$-vector, toric $f$- and $g$-polynomials of hook-shaped lattice path matroid polytopes, connecting combinatorial structures with algebraic invariants.

Contribution

It introduces explicit formulas for key polynomials of hook-shaped lattice path matroid polytopes, expanding understanding of their combinatorial and geometric properties.

Findings

Formulas for $f$-vector, toric $f$- and $g$-polynomials for hook shape cases

Connection between lattice path matroid polytopes and product of simplices

Enhanced understanding of polytope invariants in specific geometric configurations

Abstract

It is known that a lattice path matroid polytope can be associated with two given noncrossing lattice paths on $\mathbb{Z}\times\mathbb{Z}$ with the same end points. In this short note we give explicit formulae for the $f$-vector, toric $f$- and $g$-polynomials of a lattice path matroid polytope when two boundary paths enclose a hook shape.